Hi,
I am looking for a way to stream my 300GB itunes music collection externally so that I can listen to music while at work. The music is on my MBP, but I would rather find a solution that does not require me to keep my MBP on all day.

I think I need some kind of network drive : the music would be backed up on it (wirelessly and seamlessly through TimeMachine if possible) and would stream from it, either to my iPhone with an app (?) or to my office computer's iTunes.

If the OP has 300 GB of music, s/he has far exceeded iTunes Match's 25,000 song limit unless all of the music is stored as wavs of aiffs.

Plex and its myPlex feature will do you what you want. However, in order to leave your computer off and still be able to stream you will need to store your collection on a NAS device which is capable of running the Plex server software. AirPorts won't work.